Universality of Logarithmic Loss in Lossy Compression

TL;DR

This paper demonstrates the universal applicability of logarithmic loss as a distortion measure in lossy compression, showing equivalence to arbitrary distortion criteria and successively refinable properties for sources.

Contribution

It proves the universality of logarithmic loss in lossy compression and its role in successively refinable source coding under arbitrary distortion measures.

Findings

Logarithmic loss is equivalent to any distortion criterion in fixed-length lossy compression.

Sources are successively refinable under logarithmic loss for the first decoder.

Any discrete memoryless source can be successively refined under arbitrary distortion with a logarithmic loss first decoder.

Abstract

We establish two strong senses of universality of logarithmic loss as a distortion criterion in lossy compression: For any fixed length lossy compression problem under an arbitrary distortion criterion, we show that there is an equivalent lossy compression problem under logarithmic loss. In the successive refinement problem, if the first decoder operates under logarithmic loss, we show that any discrete memoryless source is successively refinable under an arbitrary distortion criterion for the second decoder.